Embedded newform 335.2.w.a.2.20

• Label: 335.2.w.a.2.20
• Level: $335$
• Weight: $2$
• Character: 335.2
• Analytic conductor: $2.675$
• Analytic rank: $0$
• Dimension: $1280$
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 335 = 5 \cdot 67 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	335.w (of order \(132\), degree \(40\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(2.67498846771\)
Analytic rank:	\(0\)
Dimension:	\(1280\)
Relative dimension:	\(32\) over \(\Q(\zeta_{132})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{132}]$

Embedding invariants

Embedding label			2.20
Character	\(\chi\)	\(=\)	335.2
Dual form			335.2.w.a.168.20

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.418693 + 0.460578i) q^{2} +(-0.107784 - 0.197392i) q^{3} +(0.153284 - 1.60526i) q^{4} +(-1.35831 + 1.77623i) q^{5} +(0.0457858 - 0.132289i) q^{6} +(-0.0559963 + 0.175172i) q^{7} +(1.80011 - 1.34755i) q^{8} +(1.59458 - 2.48121i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 1280 q - 38 q^{2} - 44 q^{3} - 44 q^{5} - 80 q^{6} - 38 q^{7} - 110 q^{8}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/335\mathbb{Z}\right)^\times\).

\(n\)	\(136\)	\(202\)
\(\chi(n)\)	\(e\left(\frac{1}{66}\right)\)	\(e\left(\frac{1}{4}\right)\)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).

(See \(a_n\) instead)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.418693	+	0.460578i	0.296061	+	0.325678i	0.869842	−	0.493331i	\(-0.164221\pi\)
							−0.573781	+	0.819009i	\(0.694524\pi\)
\(3\)	−0.107784	−	0.197392i	−0.0622291	−	0.113964i	0.844713	−	0.535220i	\(-0.179771\pi\)
							−0.906942	+	0.421256i	\(0.861589\pi\)
\(5\)	−1.35831	+	1.77623i	−0.607455	+	0.794354i
							\(7\)	−0.0559963	+	0.175172i	−0.0211646	+	0.0662086i	−0.962524	−	0.271198i	\(-0.912580\pi\)
0.941359	+	0.337407i	\(0.109550\pi\)
\(11\)	4.51845	−	1.56385i	1.36236	−	0.471519i	0.454486	−	0.890754i	\(-0.349823\pi\)
							0.907878	+	0.419235i	\(0.137702\pi\)
\(13\)	1.09735	+	0.131204i	0.304350	+	0.0363895i	0.269492	−	0.963003i	\(-0.413144\pi\)
							0.0348586	+	0.999392i	\(0.488902\pi\)
\(17\)	3.14473	+	2.59650i	0.762708	+	0.629745i	0.934734	−	0.355347i	\(-0.115637\pi\)
							−0.172026	+	0.985092i	\(0.555031\pi\)
\(19\)	1.28965	−	2.50157i	0.295866	−	0.573899i	−0.693309	−	0.720640i	\(-0.743848\pi\)
							0.989175	+	0.146741i	\(0.0468784\pi\)
\(23\)	−1.71803	−	1.04719i	−0.358234	−	0.218355i	0.329771	−	0.944061i	\(-0.393028\pi\)
							−0.688005	+	0.725706i	\(0.741513\pi\)
\(29\)	−0.438940	−	0.253422i	−0.0815091	−	0.0470593i	0.458692	−	0.888596i	\(-0.348318\pi\)
							−0.540201	+	0.841536i	\(0.681652\pi\)
\(31\)	−2.25901	+	2.87257i	−0.405731	+	0.515929i	−0.945384	−	0.325959i	\(-0.894313\pi\)
							0.539653	+	0.841887i	\(0.318555\pi\)
\(37\)	−1.25199	+	4.67251i	−0.205827	+	0.768156i	0.783369	+	0.621557i	\(0.213499\pi\)
							−0.989196	+	0.146599i	\(0.953167\pi\)
\(41\)	−2.83580	+	2.01936i	−0.442878	+	0.315372i	−0.779651	−	0.626214i	\(-0.784604\pi\)
							0.336774	+	0.941586i	\(0.390664\pi\)
\(43\)	−6.54213	+	2.44009i	−0.997666	+	0.372110i	−0.794652	−	0.607066i	\(-0.792347\pi\)
							−0.203014	+	0.979176i	\(0.565074\pi\)
\(47\)	5.69100	+	0.135471i	0.830118	+	0.0197605i	0.436709	−	0.899603i	\(-0.356144\pi\)
							0.393409	+	0.919364i	\(0.371296\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	335.2.w.a.2.20	✓	1280
5.3	odd	4	inner	335.2.w.a.203.20	yes	1280
67.34	odd	66	inner	335.2.w.a.302.20	yes	1280
335.168	even	132	inner	335.2.w.a.168.20	yes	1280

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
335.2.w.a.2.20	✓	1280	1.1	even	1	trivial
335.2.w.a.168.20	yes	1280	335.168	even	132	inner
335.2.w.a.203.20	yes	1280	5.3	odd	4	inner
335.2.w.a.302.20	yes	1280	67.34	odd	66	inner